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The Amplituhedron is a subspace of the Grassmannian that was recently defined by Arkani-Hamed and Trnka in their study of scattering amplitudes in planar $\mathcal{N}=4$ super Yang Mills theory (arXiv:1312.2007), and was the subject of many papers in the last decade. In this work we define a tropical analog of the amplituhedron, and develop techniques to address it. We prove that many of the key properties of the amplituhedron hold also in this simpler, piecewise linear, model., Comment: 44 pages, 2 figures

The papers [5, 3, 6, 19, 20] initiated the study of open $r$-spin and open FJRW intersection theories, and related them to integrable hierarchies and mirror symmetry. This paper uses a new technique, the point insertion technique, developed in the prequel [36], to define new open r-spin and open FJRW intersection theories. These new constructions provide potential candidates for theories whose existence was conjectured before: $\bullet$ K. Hori [23] predicted the existence of open $r$-spin theory with $\lfloor\frac{r}{2}\rfloor$ types of boundary states. The one constructed in [5, 3] has only one type of boundary state. In this work we describe $\lfloor\frac{r}{2}\rfloor$ open $r$-spin theories, labelled by $\mathfrak{h}\in\{0,\ldots,\lfloor\frac{r}{2}\rfloor-1\},$ where the $\mathfrak{h}$-th one has $\mathfrak{h}+1$ boundary states. We prove that the $\mathfrak{h}=0$ theory is equivalent to the [5, 3] construction, and calculate all intersection numbers for all these theories. $\bullet$ In [1] K. Aleshkin and C.C.M. Liu conjectured the existence of a quintic Fermat FJRW theory. We construct such an FJRW theory, and provide evidence that this is the conjectured theory. We also explain how the point insertion technique can be used for constructing other open enumerative theories, satisfying the same universal recursions.

The papers [3, 1, 4, 10] constructed an intersection theory on the moduli space of $r$-spin disks, and proved it satisfies mirror symmetry and relations with integrable hierarchies. That theory considered only disks with a single type of boundary state. In this work, we initiate the study of more general $r$-spin surfaces. We define the notion of graded $r$-spin surfaces with multiple internal and boundary states, and their moduli spaces. In $g=0,$ the disk case, we also define the associated open Witten bundle, and prove that the Witten bundle is canonically oriented relative to the moduli space. Moreover, we describe a method for gluing several moduli spaces along certain boundaries, show that gluing lifts to the Witten bundle and relative cotangent line bundles, and that the result is again canonically relatively oriented. In the sequel [20], we construct, based on the work of this paper, a family of $\lfloor r/2\rfloor$ intersection theories indexed by $\mathfrak{h}\in\{0,\ldots,\lfloor r/2\rfloor-1\},$ where the $\mathfrak{h}$-th one has $\mathfrak{h}+1$ possible boundary states, and calculate their intersection numbers. The $\mathfrak{h}=0$ theory is equivalent to the one constructed in [3, 1].

In our previous two papers, we constructed an $r$-spin theory in genus zero for Riemann surfaces with boundary and fully determined the corresponding intersection numbers, providing an analogue of Witten's $r$-spin conjecture in genus zero in the open setting. In particular, we proved that the generating series of open $r$-spin intersection numbers is determined by the genus-zero part of a special solution of a certain extension of the Gelfand-Dickey hierarchy, and we conjectured that the whole solution controls the open $r$-spin intersection numbers in all genera, which do not yet have a geometric definition. In this paper, we provide geometric and algebraic evidence for the correctness of this conjecture., Comment: The content of this paper was included in the content of the first versions (v1-v4) of arXiv:1809.02536, before we splitted it. It includes a discussion of the structure of open r-spin intersection numbers in higher genera. 13 pages

We construct an open enumerative theory for the Landau-Ginzburg (LG) model $(\mathbb{C}^2, \mu_r\times \mu_s, x^r+y^s)$. The invariants are defined as integrals of multisections of a Witten bundle with descendents over a moduli space that is a real orbifold with corners. In turn, a generating function for these open invariants yields the mirror LG model and a versal deformation of it with flat coordinates. After establishing an open topological recursion result, we prove an LG/LG open mirror symmetry theorem in dimension two with all descendents. The open invariants we define are not unique but depend on boundary conditions that, when altered, exhibit wall-crossing phenomena for the invariants. We describe an LG wall-crossing group classifying the wall-crossing transformations that can occur., Comment: 140 pages, submitted version

We show that a generating function for open $r$-spin enumerative invariants produces a universal unfolding of the polynomial $x^r$. Further, the coordinates parametrizing this universal unfolding are flat coordinates on the Frobenius manifold associated to the Landau-Ginzburg model $(\mathbb{C},x^r)$ via Saito-Givental theory. This result provides evidence for the same phenomenon to occur in higher dimension, proven in a sequel paper., Comment: 11 pages

The amplituhedron Ank4 is a geometric object, introduced by Arkani-Hamed and Trnka (2013) in the study of scattering amplitudes in quantum field theories. They conjecture that Ank4 admits a decomposition into images of BCFW positroid cells, arising from the Britto--Cachazo--Feng--Witten recurrence (2005). We prove that this conjecture is true.

Local to global machinery plays an important role in the study of simplicial complexes, since the seminal work of Garland [G] to our days. In this work we develop a local to global machinery for general posets. We show that the high dimensional expansion notions and many recent expansion results have a generalization to posets. Examples are fast convergence of high dimensional random walks generalizing [KO,AL], an equivalence with a global random walk definition, generalizing [DDFH] and a trickling down theorem, generalizing [O]. In particular, we show that some posets, such as the Grassmannian poset, exhibit qualitatively stronger trickling down effect than simplicial complexes. Using these methods, and the novel idea of Posetification, to Ramanujan complexes [LSV1,LSV2], we construct a constant degree expanding Grassmannian poset, and analyze its expansion. This it the first construction of such object, whose existence was conjectured in [DDFH]., Comment: Changed the title; Changed some notations; Simplified the notion of 2-skeleton regularity and modified the proof of Lemma 5.4 to be compatible with the simplification

Given a random text over a finite alphabet, we study the frequencies at which fixed-length words occur as subsequences. As the data size grows, the joint distribution of word counts exhibits a rich asymptotic structure. We investigate all linear combinations of subword statistics, and fully characterize their different orders of magnitude using diverse algebraic tools. Moreover, we establish the spectral decomposition of the space of word statistics of each order. We provide explicit formulas for the eigenvectors and eigenvalues of the covariance matrix of the multivariate distribution of these statistics. Our techniques include and elaborate on a set of algebraic word operators, recently studied and employed by Dieker and Saliola (Adv Math, 2018). Subword counts find applications in Combinatorics, Statistics, and Computer Science. We revisit special cases from the combinatorial literature, such as intransitive dice, random core partitions, and questions on random walk. Our structural approach describes in a unified framework several classical statistical tests. We propose further potential applications to data analysis and machine learning.

In this work we introduce a new notion of expansion in higher dimensions that is stronger than the well studied cosystolic expansion notion, and is termed {\em Collective-cosystolic expansion}. We show that tensoring two cosystolic expanders yields a new cosystolic expander, assuming one of the complexes in the product, is not only cosystolic expander, but rather a collective cosystolic expander. We then show that the well known bounded degree cosystolic expanders, the Ramanujan complexes are, in fact, collective cosystolic expanders. This enables us to construct new bounded degree cosystolic expanders, by tensoring of Ramanujan complexes. Using our new constructed bounded degree cosystolic expanders we construct {\em explicit} quantum LDPC codes of distance $\sqrt{n} \log^k n$ for any $k$, improving a recent result of Evra et. al. \cite{EKZ}, and setting a new record for distance of explicit quantum LDPC codes. The work of \cite{EKZ} took advantage of the high dimensional expansion notion known as cosystolic expansion, that occurs in Ramanujan complexes. Our improvement is achieved by considering tensor product of Ramanujan complexes, and using their newly derived property, the collective cosystolic expansion., Comment: Added many new results: In the first version it was proven that the product of two cosystolic expanders has linear cosystoles. In this version it is proven that if one of them has an additional property, then their product is itself a cosystolic expander. It is also proven that many well known cosystolic expanders have this additional property

In standard rounding, we want to map each value $X$ in a large continuous space (e.g., $R$) to a nearby point $P$ from a discrete subset (e.g., $Z$). This process seems to be inherently discontinuous in the sense that two consecutive noisy measurements $X_1$ and $X_2$ of the same value may be extremely close to each other and yet they can be rounded to different points $P_1\ne P_2$, which is undesirable in many applications. In this paper we show how to make the rounding process perfectly continuous in the sense that it maps any pair of sufficiently close measurements to the same point. We call such a process consistent rounding, and make it possible by allowing a small amount of information about the first measurement $X_1$ to be unidirectionally communicated to and used by the rounding process of $X_2$. The fault tolerance of a consistent rounding scheme is defined by the maximum distance between pairs of measurements which guarantees that they are always rounded to the same point, and our goal is to study the possible tradeoffs between the amount of information provided and the achievable fault tolerance for various types of spaces. When the measurements $X_i$ are arbitrary vectors in $R^d$, we show that communicating $\log_2(d+1)$ bits of information is both sufficient and necessary (in the worst case) in order to achieve consistent rounding for some positive fault tolerance, and when d=3 we obtain a tight upper and lower asymptotic bound of $(0.561+o(1))k^{1/3}$ on the achievable fault tolerance when we reveal $\log_2(k)$ bits of information about how $X_1$ was rounded. We analyze the problem by considering the possible colored tilings of the space with $k$ available colors, and obtain our upper and lower bounds with a variety of mathematical techniques including isoperimetric inequalities, the Brunn-Minkowski theorem, sphere packing bounds, and \v{C}ech cohomology.

We lay the foundation for a version of $r$-spin theory in genus zero for Riemann surfaces with boundary. In particular, we define the notion of $r$-spin disks, their moduli space, and the Witten bundle, we show that the moduli space is a compact smooth orientable orbifold with corners, and we prove that the Witten bundle is canonically relatively oriented relative to the moduli space. In the sequel to this paper, we use these constructions to define open $r$-spin intersection theory and relate it to the Gelfand-Dickey hierarchy, thus providing an analogue of Witten's $r$-spin conjecture in the open setting., Comment: The content of this paper was included in the content of the first versions (v1-v3) of arXiv:1809.02536, before we splitted it. It includes the definition of r-spin disks, the construction of the moduli spaces and bundles, and the proofs of orientation theorems

We define stationary descendent integrals on the moduli space of stable maps from disks to $(\mathbb{CP}^1,\mathbb{RP}^1)$. We prove a localization formula for the stationary theory involving contributions from the fixed points and from all the corner-strata. We use the localization formula to prove a recursion relation and a closed formula for all genus $0$ disk cover invariants in the stationary case. For all higher genus invariants, we propose a conjectural formula., Comment: Minor changes in v3: affiliations, grant acknowledgements, and references

We conclude the construction of $r$-spin theory in genus zero for Riemann surfaces with boundary. In particular, we define open $r$-spin intersection numbers, and we prove that their generating function is closely related to the wave function of the $r$th Gelfand-Dickey integrable hierarchy. This provides an analogue of Witten's $r$-spin conjecture in the open setting and a first step toward the construction of an open version of Fan-Jarvis-Ruan-Witten theory. As an unexpected consequence, we establish a mysterious relationship between open $r$-spin theory and an extension of Witten's closed theory., Comment: v5: Comparing to the previous version, v4, the results about open $r$-spin intersection numbers in higher genera were moved to the article arXiv:2211.16302. Accepted for publication in the Journal of Differential Geometry. Comparing to the accepted version, contains Remarks 2.2, 3.6, and 4.9. 63 pages

We prove a weighted generalization of the formula for the number of plane vertex-labeled trees.

We study a generalization of genus-zero $r$-spin theory in which exactly one insertion has a negative-one twist, which we refer to as the "closed extended" theory, and which is closely related to the open $r$-spin theory of Riemann surfaces with boundary. We prove that the generating function of genus-zero closed extended intersection numbers coincides with the genus-zero part of a special solution to the system of differential equations for the wave function of the $r$-th Gelfand-Dickey hierarchy. This parallels an analogous result for the open $r$-spin generating function in the companion paper to this work., Comment: v3: 25 pages, minor corrections according to referee's remarks

A study of the intersection theory on the moduli space of Riemann surfaces with boundary was recently initiated in a work of R. Pandharipande, J. P. Solomon and the third author, where they introduced open intersection numbers in genus 0. Their construction was later generalized to all genera by J. P. Solomon and the third author. In this paper we consider a refinement of the open intersection numbers by distinguishing contributions from surfaces with different numbers of boundary components, and we calculate all these numbers. We then construct a matrix model for the generating series of the refined open intersection numbers and conjecture that it is equivalent to the Kontsevich-Penner matrix model. An evidence for the conjecture is presented. Another refinement of the open intersection numbers, which describes the distribution of the boundary marked points on the boundary components, is also discussed., Comment: 35 pages, 13 figures; references added

We consider the Edwards-Anderson Ising Spin Glass model for non negative temperatures T: We define the natural notion of Boltzmann- Gibbs measure for the Edwards-Anderson spin glass at a given temperature, and of unsatisfied edges. We prove that for low enough temperatures, in almost every spin configuration the graph formed by the unsatisfied edges is made of finite connected components. In other words, the unsatisfied edges do not percolate.

A Hamiltonian cycle in a graph is a spanning subgraph that is homeomorphic to a circle. With this in mind, it is natural to define a Hamiltonian d-sphere in a d-dimensional simplicial complex as a spanning subcomplex that is homeomorphic to a d-dimensional sphere. We consider the Linial-Meshulam model for random simplicial complexes, and prove that there is a sharp threshold at $p=\sqrt{\frac{e}{\gamma n}}$ for the appearance of a Hamiltonian $2$-sphere in a random $2$-complex, where $\gamma = 4^4/3^3$., Comment: Main changes from last version: An explanation of the proof structure was added as a subsection in the introduction. A computational mistake in Lemma 4.5 was corrected, and a result some constants throughout the proof had to be modified (in the definition of good graph, in the final estimation) . Some other minor corrections

A classical longstanding open problem in statistics is finding an explicit expression for the probability measure which maximizes entropy with respect to given constraints. In this paper a solution to this problem is found, using perturbative Feynman calculus. The explicit expression is given as a sum over weighted trees., Comment: A few minor corrections

In recent works, [20],[21], descendent integrals on the moduli space of Riemann surfaces with boundary were defined. It was conjectured in [20] that the generating function of these integrals satisfies the open KdV equations. In this paper we develop the notions of symmetric Strebel-Jenkins differentials and of Kasteleyn orientations for graphs embedded in open surfaces. In addition we write an explicit expression for the angular form of the sum of line bundles. Using these tools we prove a formula for the descendent integrals in terms of sums over weighted graphs. Based on this formula, the conjecture of [20] was proved in [5]., Comment: Improved the writing of some proofs and definitions, fixed some typos. To appear in G&T

In a recent work, R. Pandharipande, J. P. Solomon and the second author have initiated a study of the intersection theory on the moduli space of Riemann surfaces with boundary. They conjectured that the generating series of the intersection numbers satisfies the open KdV equations. In this paper we prove this conjecture. Our proof goes through a matrix model and is based on a Kontsevich type combinatorial formula for the intersection numbers that was found by the second author., Comment: v5: a step in the proof of Proposition 4.1, pointed by Jan Troost, is made more clear; 27 pages

We define a theory of descendent integration on the moduli spaces of stable pointed disks. The descendent integrals are proved to be coefficients of the $\tau$-function of an open KdV heirarchy. A relation between the integrals and a representation of half the Virasoro algebra is also proved. The construction of the theory requires an in depth study of homotopy classes of multivalued boundary conditions. Geometric recursions based on the combined structure of the boundary conditions and the moduli space are used to compute the integrals. We also provide a detailed analysis of orientations. Our open KdV and Virasoro constraints uniquely specify a theory of higher genus open descendent integrals. As a result, we obtain an open analog (governing all genera) of Witten's conjectures concerning descendent integrals on the Deligne-Mumford space of stable curves., Comment: 98 pages, 5 figures; v3: minor corrections, expanded introduction, added details, updated references